Repository: Freie Universit├Ąt Berlin, Math Department

A rigorous derivation and energetics of a wave equation with fractional damping

Mielke, Alexander and Netz, Roland R. and Zendehroud, Sina (2020) A rigorous derivation and energetics of a wave equation with fractional damping. arXiv . pp. 1-20. (Submitted)

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Official URL: https://arxiv.org/abs/2004.11830

Abstract

We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally-damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy-dissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionally-damped wave equation with a time derivative of order 3/2. Keywords: bulk-interface coupling, surface waves, energy-dissipation balance, fractional derivatives, convergence of semigroups, parabolic Dirichlet-to-Neumann map. MOS: 35Q35 35Q74 74J15

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics
ID Code:2442
Deposited By: Monika Drueck
Deposited On:28 May 2020 10:39
Last Modified:28 May 2020 10:39

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